Nelson-Aalen Estimate of the Cumulative Hazard Function

From Survival Products to Hazard Sums

Recall that the Kaplan-Meier estimator is multiplicative. It estimates survival by multiplying probabilities of surviving past each event time:
${\hat{S}}_{KM}(t)={\prod }_{j:t_j\le t}\left(1-\frac{d_j}{n_j}\right)$
where $d_j$ is the number of events (deaths) at time $t_j$ and $n_j$ is the number of individuals at risk just before that time.

The Nelson-Aalen approach is, by contrast, additive. It operates on the principle that the instantaneous hazard rate at an event time $t_j$ can be estimated by the proportion of the risk set that experiences the event, i.e., $\hat{h}(t_j)=d_j/n_j$. The cumulative hazard $H(t)$ is the sum (or integral, in the continuous case) of these hazard rates over time. The NA estimator builds the cumulative hazard function by simply summing these discrete hazard contributions.

This conceptual shift from a product of probabilities to a sum of hazards provides a different perspective on the same underlying data.

The Nelson-Aalen Estimator: Formula and Calculation

The formal definition of the Nelson-Aalen estimator is straightforward. It starts at zero, $\hat{H}(0)=0$, and increases at each event time. The following reading from your core actuarial study manual introduces the formula and provides a worked example.

As the reading makes clear, the formula is:
${\hat{H}}_{NA}(t)={\sum }_{j:t_j\le t}\frac{d_j}{n_j}$
Visually, this produces a right-continuous step function. It is constant between events and jumps upwards at each event time $t_j$ by an amount equal to the estimated hazard, $d_j/n_j$.

This non-parametric, data-driven step function can be contrasted with the smooth, pre-defined shape of a parametric cumulative hazard function, such as the straight line $H(t)=\lambda t$ that arises from an exponential survival model.

A Worked Example

Let's apply the formula to a small dataset to ensure the calculation is clear.
Suppose we start a study with 5 individuals. The observed times are: 2, 5, 5+, 8, 9+. The '+' indicates a right-censored observation.

The event times are $t=2$ and $t=8$.

1. At time $t=2$:

   • The number at risk, $n_1$, is all 5 individuals.
   • The number of events, $d_1$, is 1.
   • The cumulative hazard is $\hat{H}(2)=\frac{d_1}{n_1}=\frac{1}{5}=0.2$.
   • For any time $t$ where $2\le t<8$, $\hat{H}(t)=0.2$.

2. At time $t=8$:

   • To find the number at risk, $n_2$, we start with the 5 individuals, subtract the 1 who had an event at $t=2$, and subtract the 1 who was censored at $t=5$. So, $n_2=5-1-1=3$.
   • The number of events, $d_2$, is 1.
   • The cumulative hazard is incremented:
     $\hat{H}(8)=\hat{H}(\text{just before 8})+\frac{d_2}{n_2}=0.2+\frac{1}{3}\approx 0.533$
   • For any time $t\ge 8$ (until the next event), $\hat{H}(t)\approx 0.533$.

Practice Problem

Now, let's try a typical exam-style problem that requires you to work backwards, testing your understanding of the estimator's sequential construction.

Solution Walkthrough:
The problem states the data were not truncated or censored and at most one claim was paid at any one time (i.e., all $d_j=1$). Let $N$ be the initial number of claims.

The NA estimate after the second claim is:
$\hat{H}(t_2)=\frac{1}{N}+\frac{1}{N-1}=\frac{23}{132}$
We need to find $N$. We can solve the quadratic, but given your background, you might quickly estimate that the two terms are roughly equal, so $2/(N-0.5)\approx 23/132$, which gives $N-0.5\approx 2\times 132/23\approx 11.48$. This suggests $N=12$. Let's check:
$\frac{1}{12}+\frac{1}{11}=\frac{11+12}{132}=\frac{23}{132}$
The value $N=12$ is correct.

Now we need the estimate immediately following the fourth paid claim. We just continue the summation:
$\hat{H}(t_4)=\hat{H}(t_2)+\frac{1}{N-2}+\frac{1}{N-3}$
$\hat{H}(t_4)=\frac{23}{132}+\frac{1}{10}+\frac{1}{9}\approx 0.1742+0.1+0.1111=0.3853$
This corresponds to answer choice (C).

Connecting Nelson-Aalen back to Survival

The primary output of the NA estimator is the cumulative hazard, but we can easily derive a survival function estimate using $\hat{S}(t)=\exp (-\hat{H}(t))$. This is sometimes called the Fleming-Harrington estimator.

An interesting theoretical point arises when comparing this NA-derived survival estimate with the Kaplan-Meier estimate. For any dataset, it can be shown that:
${\hat{S}}_{NA}(t)\ge {\hat{S}}_{KM}(t)$
The estimates are very close, but the NA-derived survival probability is always slightly higher (more optimistic) than the direct KM estimate. This is because the KM estimator uses the term $(1-d_j/n_j)$, while the NA approach effectively uses $\exp (-d_j/n_j)$. From the series expansion of the exponential function, we know that $\exp (-x)>1-x$ for any $x>0$. This subtle difference is a direct consequence of the additive versus multiplicative approaches.

Conclusion

In this lesson, we have added the Nelson-Aalen estimator to your non-parametric modeling toolkit. It provides a direct way to estimate and visualize the cumulative hazard from potentially censored data.

Key Takeaways:

• The Nelson-Aalen estimator is a non-parametric method for estimating the cumulative hazard function, $H(t)$.
• The estimate is calculated as an additive sum of the hazard contributions at each event time: $\hat{H}(t)={\sum }_{j:t_j\le t}{d}_j/n_j$.
• Like the Kaplan-Meier survival curve, the NA estimate for $H(t)$ is a right-continuous step function.
• A corresponding survival function estimate can be obtained via the relationship $\hat{S}(t)=\exp (-\hat{H}(t))$.

Having learned two distinct methods for analyzing survival in a single population, our next step is to address a fundamental question in research: how do we compare survival between different groups? In the next lesson, we will introduce the log-rank test, a widely used statistical test for formally comparing two or more survival curves.